Question: Suppose $x$ and $y$ are integers such that  $xy+5x+4y=-5$.  Find the greatest possible value of $y$.
Note that $(x+4)(y+5)$ equals $xy+5x+4y+20$.  So, add $20$ to both sides of the original equation to get $xy+5x+4y+20=15$, so now we may apply Simon's Favorite Factoring Trick and write the equation as $(x+4)(y+5)=15$.

Then, the potential ordered pairs $((x+4),(y+5))$ with $x<y$ are $(-15,-1)$, $(-5,-3)$, $(1,15)$ and $(3,5)$, since these are the pairs of integers that multiply to 15.  The greatest value for $y+5$ is thus $15$.  We solve $y+5=15$ for $y$ to yield $y=\boxed{10}$.